Question

Two rods, one made of aluminium and the other made of steel, having initial length $l_1$ and $l_2$ are connected together to form a single rod of length $l_1+l_2$. The coefficients of linear expansion for aluminium and steel are ${\alpha }_a$ and ${\alpha }_s$ respectively. If the length of each rod increases by the same amount when their temperature is raised by $t^{\circ }C$,then find the ratio $\frac{l_1}{l_1+l_2}$

• A. $\frac{{\alpha }_s}{{\alpha }_a+{\alpha }_s}$
• B. $\frac{{\alpha }_a}{{\alpha }_s}$
• C. $\frac{{\alpha }_a}{({\alpha }_a+{\alpha }_s)}$
• D. $\frac{{\alpha }_s}{{\alpha }_a}$

Answer 1

The correct option is A $\frac{{\alpha }_s}{{\alpha }_a+{\alpha }_s}$

Formula used: $\Delta l=l_o\alpha \Delta T$

Given:
$(\Delta l{)}_a=(\Delta l{)}_s$

$(\Delta l{)}_1=(\Delta l{)}_2$

$l_1{\alpha }_a\Delta \theta =l_2{\alpha }_s\Delta \theta$

$l_2=\frac{{\alpha }_a}{{\alpha }_s}{l}_1$

$\frac{l_1}{l_1+l_2}=\frac{l_1}{l_1+\frac{{\alpha }_a}{{\alpha }_s}{l}_1}=\frac{{\alpha }_s}{{\alpha }_a+{\alpha }_s}$

Final answer: (c)